Properties

Label 2013.2.a.b
Level $2013$
Weight $2$
Character orbit 2013.a
Self dual yes
Analytic conductor $16.074$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2013,2,Mod(1,2013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2013.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(16.0738859269\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 2x^{10} - 14x^{9} + 27x^{8} + 66x^{7} - 125x^{6} - 115x^{5} + 227x^{4} + 40x^{3} - 129x^{2} + 26x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_{8} q^{5} + \beta_1 q^{6} + ( - \beta_{9} - 1) q^{7} + (\beta_{7} - \beta_{6} + \beta_{4} - \beta_{2} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_{8} q^{5} + \beta_1 q^{6} + ( - \beta_{9} - 1) q^{7} + (\beta_{7} - \beta_{6} + \beta_{4} - \beta_{2} - \beta_1) q^{8} + q^{9} + (\beta_{10} - \beta_{8} - \beta_{4} + \beta_1 - 1) q^{10} + q^{11} + ( - \beta_{2} - 1) q^{12} + ( - \beta_{10} + \beta_{9} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{13} + (\beta_{9} - \beta_{7} + \beta_1) q^{14} - \beta_{8} q^{15} + ( - \beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_1) q^{16} + ( - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{17} - \beta_1 q^{18} + (\beta_{9} - \beta_{8} - \beta_{7} - \beta_{4} + \beta_1 - 2) q^{19} + ( - \beta_{9} - \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{2} + \cdots - 2) q^{20}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - 2 q^{2} - 11 q^{3} + 10 q^{4} - q^{5} + 2 q^{6} - 11 q^{7} - 3 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q - 2 q^{2} - 11 q^{3} + 10 q^{4} - q^{5} + 2 q^{6} - 11 q^{7} - 3 q^{8} + 11 q^{9} - 8 q^{10} + 11 q^{11} - 10 q^{12} - 13 q^{13} + 5 q^{14} + q^{15} + 4 q^{16} - 13 q^{17} - 2 q^{18} - 12 q^{19} - 7 q^{20} + 11 q^{21} - 2 q^{22} - 3 q^{23} + 3 q^{24} + 12 q^{25} + 12 q^{26} - 11 q^{27} - 13 q^{28} + 2 q^{29} + 8 q^{30} + q^{31} - 23 q^{32} - 11 q^{33} - 14 q^{34} - 4 q^{35} + 10 q^{36} - 14 q^{37} - 8 q^{38} + 13 q^{39} - 34 q^{40} + 3 q^{41} - 5 q^{42} - 21 q^{43} + 10 q^{44} - q^{45} - 12 q^{46} - 16 q^{47} - 4 q^{48} - 18 q^{49} - 13 q^{50} + 13 q^{51} - 33 q^{52} + 2 q^{54} - q^{55} + 16 q^{56} + 12 q^{57} - 17 q^{58} + 3 q^{59} + 7 q^{60} + 11 q^{61} - 21 q^{62} - 11 q^{63} - 7 q^{64} - q^{65} + 2 q^{66} - 24 q^{67} + 2 q^{68} + 3 q^{69} + 4 q^{70} + 7 q^{71} - 3 q^{72} - 42 q^{73} - 16 q^{74} - 12 q^{75} - 13 q^{76} - 11 q^{77} - 12 q^{78} - 11 q^{79} + 42 q^{80} + 11 q^{81} - 38 q^{82} - 34 q^{83} + 13 q^{84} - 14 q^{85} + 42 q^{86} - 2 q^{87} - 3 q^{88} + 29 q^{89} - 8 q^{90} + 9 q^{91} + 42 q^{92} - q^{93} - 33 q^{94} - 31 q^{95} + 23 q^{96} - 45 q^{97} - 33 q^{98} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 2x^{10} - 14x^{9} + 27x^{8} + 66x^{7} - 125x^{6} - 115x^{5} + 227x^{4} + 40x^{3} - 129x^{2} + 26x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6 \nu^{10} - 3 \nu^{9} - 80 \nu^{8} + 42 \nu^{7} + 340 \nu^{6} - 257 \nu^{5} - 506 \nu^{4} + 688 \nu^{3} + 167 \nu^{2} - 532 \nu + 72 ) / 17 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7 \nu^{10} + 5 \nu^{9} - 99 \nu^{8} - 70 \nu^{7} + 459 \nu^{6} + 264 \nu^{5} - 834 \nu^{4} - 257 \nu^{3} + 583 \nu^{2} + 48 \nu - 86 ) / 17 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{10} - 9 \nu^{9} - 19 \nu^{8} + 126 \nu^{7} + 136 \nu^{6} - 584 \nu^{5} - 379 \nu^{4} + 1061 \nu^{3} + 280 \nu^{2} - 678 \nu + 46 ) / 17 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13 \nu^{10} - 2 \nu^{9} + 179 \nu^{8} + 45 \nu^{7} - 799 \nu^{6} - 211 \nu^{5} + 1323 \nu^{4} + 266 \nu^{3} - 699 \nu^{2} - 111 \nu + 31 ) / 17 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 20 \nu^{10} - 7 \nu^{9} + 278 \nu^{8} + 115 \nu^{7} - 1258 \nu^{6} - 475 \nu^{5} + 2157 \nu^{4} + 506 \nu^{3} - 1265 \nu^{2} - 74 \nu + 66 ) / 17 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 8 \nu^{10} - 21 \nu^{9} - 118 \nu^{8} + 277 \nu^{7} + 612 \nu^{6} - 1204 \nu^{5} - 1247 \nu^{4} + 1960 \nu^{3} + 676 \nu^{2} - 1004 \nu + 113 ) / 17 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 23 \nu^{10} - 3 \nu^{9} - 318 \nu^{8} + 25 \nu^{7} + 1428 \nu^{6} - 189 \nu^{5} - 2359 \nu^{4} + 705 \nu^{3} + 1119 \nu^{2} - 583 \nu + 89 ) / 17 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 20 \nu^{10} - 10 \nu^{9} - 278 \nu^{8} + 123 \nu^{7} + 1275 \nu^{6} - 613 \nu^{5} - 2225 \nu^{4} + 1347 \nu^{3} + 1231 \nu^{2} - 878 \nu + 36 ) / 17 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + 6\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{8} - 9\beta_{7} + 10\beta_{6} - 2\beta_{5} - 8\beta_{4} + 9\beta_{2} + 28\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 9 \beta_{10} - 2 \beta_{9} + 11 \beta_{8} - 13 \beta_{7} + 13 \beta_{6} - 12 \beta_{5} - \beta_{4} + 11 \beta_{3} + 37 \beta_{2} + 13 \beta _1 + 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{10} + 13 \beta_{8} - 68 \beta_{7} + 81 \beta_{6} - 25 \beta_{5} - 54 \beta_{4} + 2 \beta_{3} + 70 \beta_{2} + 167 \beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 65 \beta_{10} - 25 \beta_{9} + 92 \beta_{8} - 123 \beta_{7} + 125 \beta_{6} - 108 \beta_{5} - 20 \beta_{4} + 92 \beta_{3} + 242 \beta_{2} + 128 \beta _1 + 434 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 20 \beta_{10} - 2 \beta_{9} + 125 \beta_{8} - 495 \beta_{7} + 612 \beta_{6} - 230 \beta_{5} - 354 \beta_{4} + 35 \beta_{3} + 524 \beta_{2} + 1044 \beta _1 + 283 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 444 \beta_{10} - 221 \beta_{9} + 702 \beta_{8} - 1030 \beta_{7} + 1067 \beta_{6} - 870 \beta_{5} - 237 \beta_{4} + 694 \beta_{3} + 1651 \beta_{2} + 1122 \beta _1 + 2653 \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.70023
2.05394
2.04468
1.21038
0.942842
0.175402
0.0512060
−1.03852
−1.65258
−2.09663
−2.39095
−2.70023 −1.00000 5.29122 2.21957 2.70023 −2.11895 −8.88702 1.00000 −5.99333
1.2 −2.05394 −1.00000 2.21866 −0.0248942 2.05394 1.52855 −0.449108 1.00000 0.0511312
1.3 −2.04468 −1.00000 2.18073 −1.44717 2.04468 −4.07149 −0.369530 1.00000 2.95900
1.4 −1.21038 −1.00000 −0.534974 −4.03730 1.21038 0.314382 3.06829 1.00000 4.88668
1.5 −0.942842 −1.00000 −1.11105 3.49194 0.942842 1.81837 2.93323 1.00000 −3.29234
1.6 −0.175402 −1.00000 −1.96923 −1.94648 0.175402 −2.33819 0.696212 1.00000 0.341417
1.7 −0.0512060 −1.00000 −1.99738 3.74211 0.0512060 −4.65643 0.204690 1.00000 −0.191618
1.8 1.03852 −1.00000 −0.921477 −1.07279 −1.03852 1.21131 −3.03401 1.00000 −1.11412
1.9 1.65258 −1.00000 0.731020 2.07534 −1.65258 −0.262572 −2.09709 1.00000 3.42966
1.10 2.09663 −1.00000 2.39586 −1.65820 −2.09663 −1.74412 0.829977 1.00000 −3.47663
1.11 2.39095 −1.00000 3.71663 −2.34211 −2.39095 −0.680854 4.10437 1.00000 −5.59986
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(-1\)
\(61\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2013.2.a.b 11
3.b odd 2 1 6039.2.a.c 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.b 11 1.a even 1 1 trivial
6039.2.a.c 11 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{11} + 2 T_{2}^{10} - 14 T_{2}^{9} - 27 T_{2}^{8} + 66 T_{2}^{7} + 125 T_{2}^{6} - 115 T_{2}^{5} - 227 T_{2}^{4} + 40 T_{2}^{3} + 129 T_{2}^{2} + 26 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2013))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} + 2 T^{10} - 14 T^{9} - 27 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{11} \) Copy content Toggle raw display
$5$ \( T^{11} + T^{10} - 33 T^{9} - 40 T^{8} + \cdots + 71 \) Copy content Toggle raw display
$7$ \( T^{11} + 11 T^{10} + 31 T^{9} - 41 T^{8} + \cdots + 31 \) Copy content Toggle raw display
$11$ \( (T - 1)^{11} \) Copy content Toggle raw display
$13$ \( T^{11} + 13 T^{10} + 7 T^{9} - 493 T^{8} + \cdots + 631 \) Copy content Toggle raw display
$17$ \( T^{11} + 13 T^{10} - 31 T^{9} + \cdots + 626503 \) Copy content Toggle raw display
$19$ \( T^{11} + 12 T^{10} - 17 T^{9} + \cdots - 10591 \) Copy content Toggle raw display
$23$ \( T^{11} + 3 T^{10} - 108 T^{9} + \cdots - 28177 \) Copy content Toggle raw display
$29$ \( T^{11} - 2 T^{10} - 230 T^{9} + \cdots + 8795203 \) Copy content Toggle raw display
$31$ \( T^{11} - T^{10} - 119 T^{9} + 80 T^{8} + \cdots + 1679 \) Copy content Toggle raw display
$37$ \( T^{11} + 14 T^{10} - 116 T^{9} + \cdots - 7494199 \) Copy content Toggle raw display
$41$ \( T^{11} - 3 T^{10} - 251 T^{9} + \cdots + 337149 \) Copy content Toggle raw display
$43$ \( T^{11} + 21 T^{10} - 62 T^{9} + \cdots + 512533 \) Copy content Toggle raw display
$47$ \( T^{11} + 16 T^{10} + \cdots - 117544331 \) Copy content Toggle raw display
$53$ \( T^{11} - 412 T^{9} + \cdots - 508890247 \) Copy content Toggle raw display
$59$ \( T^{11} - 3 T^{10} - 145 T^{9} + \cdots + 228413 \) Copy content Toggle raw display
$61$ \( (T - 1)^{11} \) Copy content Toggle raw display
$67$ \( T^{11} + 24 T^{10} + 9 T^{9} + \cdots + 15755917 \) Copy content Toggle raw display
$71$ \( T^{11} - 7 T^{10} - 358 T^{9} + \cdots - 786742397 \) Copy content Toggle raw display
$73$ \( T^{11} + 42 T^{10} + 419 T^{9} + \cdots - 10219489 \) Copy content Toggle raw display
$79$ \( T^{11} + 11 T^{10} - 179 T^{9} + \cdots - 7961 \) Copy content Toggle raw display
$83$ \( T^{11} + 34 T^{10} + 251 T^{9} + \cdots - 3176377 \) Copy content Toggle raw display
$89$ \( T^{11} - 29 T^{10} + \cdots - 371355157 \) Copy content Toggle raw display
$97$ \( T^{11} + 45 T^{10} + \cdots + 5102744443 \) Copy content Toggle raw display
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